Integer Programming and Combinatorial Optimization
Volume 1610 of the series Lecture Notes in Computer Science pp 137150
Bounds on the Chvátal Rank of Polytopes in the 0/1Cube
 Friedrich EisenbrandAffiliated withMaxPlanckInstitut für Informatik, Im Stadtwald
 , Andreas S. SchulzAffiliated withSloan School of Management and Operations Research Center, E53361, MIT
Abstract
Gomory’s and Chvátal’s cuttingplane procedure proves recursively the validity of linear inequalities for the integer hull of a given polyhedron. The number of rounds needed to obtain all valid inequalities is known as the Chvátal rank of the polyhedron. It is wellknown that the Chvátal rank can be arbitrarily large, even if the polyhedron is bounded, if it is of dimension 2, and if its integer hull is a 0/1polytope. We prove that the Chvátal rank of polyhedra featured in common relaxations of many combinatorial optimization problems is rather small; in fact, the rank of any polytope contained in the ndimensional 0/1cube is at most 3n ^{2} lg n. This improves upon a recent result of Bockmayr et al. [6] who obtained an upper bound of O(n ^{3} lg n).
Moreover, we refine this result by showing that the rank of any polytope in the 0/1cube that is defined by inequalities with small coefficients is O(n). The latter observation explains why for most cutting planes derived in polyhedral studies of several popular combinatorial optimization problems only linear growth has been observed (see, e.g., [13]); the coefficients of the corresponding inequalities are usually small. Similar results were only known for monotone polyhedra before.
Finally, we provide a family of polytopes contained in the 0/1cube the Chvátal rank of which is at least (1+∈)n for some ∈ > 0; the best known lower bound was n.
 Title
 Bounds on the Chvátal Rank of Polytopes in the 0/1Cube
 Book Title
 Integer Programming and Combinatorial Optimization
 Book Subtitle
 7th International IPCO Conference Graz, Austria, June 9–11, 1999 Proceedings
 Pages
 pp 137150
 Copyright
 1999
 DOI
 10.1007/3540487778_11
 Print ISBN
 9783540660194
 Online ISBN
 9783540487777
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 1610
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Gérard Cornuéjols ^{(4)}
 Rainer E. Burkard ^{(5)}
 Gerhard J. Woeginger ^{(5)}
 Editor Affiliations

 4. GSIA, Carnegie Mellon University
 5. Institut für Mathematik, Technische Universität Graz
 Authors

 Friedrich Eisenbrand ^{(6)}
 Andreas S. Schulz ^{(7)}
 Author Affiliations

 6. MaxPlanckInstitut für Informatik, Im Stadtwald, D66123, Saarbrücken, Germany
 7. Sloan School of Management and Operations Research Center, E53361, MIT, Cambridge, MA, 02139, USA
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